1. **Problem:** Solve the integral that leads to the expression $\ln(|\ln|x||) + c$. 2. **Formula and rules:** The integral involves nested logarithms. Recall that $\frac{d}{dx} \ln|f(x)| = \frac{f'(x)}{f(x)}$ and the chain rule applies. 3. **Intermediate work:** Let $u = \ln|x|$, then $du = \frac{1}{x} dx$. The integral becomes $\int \frac{1}{u} du = \ln|u| + c = \ln|\ln|x|| + c$. 4. **Explanation:** We substituted the inner logarithm to simplify the integral, then integrated $1/u$ which is $\ln|u|$, and substituted back. 5. **Final answer:** $$\int \frac{1}{x \ln|x|} dx = \ln|\ln|x|| + c$$